Stability, Uniqueness and Physical Inviscid Limit for Solutions of Hyperbolic Conservation Laws and Compressible Euler Equations

The compressible Euler system, first introduced in the eighteenth century, is among the oldest partial differential equation models. It is extensively used in physics and engineering, for instance, to model gas dynamics. This system serves as a fundamental model to study shock waves in gas dynamics. However, many major theoretical problems for the Euler equations, relevant to their applications, remain unresolved. This project provides insights into two of these open problems: the formation of shock waves and the uniqueness and stability of solutions that develop shock waves. The project promotes a fundamental understanding of gas dynamics while offering research-related training opportunities for undergraduate and graduate students. In this project, the investigator addresses a fundamental question for compressible Euler equations: How do solutions behave beyond the formation of a singularity, such as a shock wave? The first goal of the project is to study the stability of solutions of the compressible Euler equations, including shock waves. A second goal is to understand the inviscid limit from the Navier-Stokes equation to the Euler equations, which also provides a framework for the uniqueness of solutions for Euler equations through the inviscid limit. Another direction of research is related to the formation of shock waves and shock-free solutions. The work of the project involves both analytical and numerical techniques. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.